From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models
Résumé
In this work, we derive first order continuum traffic flow models from a microscopic delayed follow-the-leader model. Those are applicable in the context of vehicular traffic flow as well as pedestrian traffic flow. The microscopic model is based on an optimal velocity function and a reaction time parameter. The corresponding macroscopic formulations in Eulerian or Lagrangian coordinates result in first order convection-diffusion equations. More precisely, the convection is described by the optimal velocity while the diffusion term depends on the reaction time. A linear stability analysis for homogeneous solutions of both continuous and discrete models are provided. The conditions match the ones of the car-following model for specific values of the space discretization. The behavior of the novel model is illustrated thanks to numerical simulations. Transitions to collision-free self-sustained stop-and-go dynamics are obtained if the reaction time is sufficiently large. The results show that the dynamics of the microscopic model can be well captured by the macroscopic equations. For non–zero reaction times we observe a scattered fundamental diagram. The scattering width is compared to real pedestrian and road traffic data.
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